The Rees algebra R(I) = R[It] of an ideal I of a Noetherian local ring R plays a major role in commutative algebra and in algebraic geometry, since Proj(R(I)) is the blowup of the affine scheme Spec(R) along the closed subscheme Spec(R/I).
So far, the problem of describing the equations of Rees algebras of ideals, as well as other related algebras, has shown to be relevant in order to further understand these major algebraic objects. The equations of R(I) arise as the elements in the kernel Q of a symmetric presentation of R(I). While this kernel may differ from one presentation to another, the degrees of a minimal generating set of homogeneous elements are known to be independent from the chosen presentation. The top degree among such generating sets, known as the relation type and denoted by rt(I), is a coarse measurement of the complexity of the underlying Rees algebra which is nonetheless a useful numerical invariant. The ideals I such that rt(I) = 1, known as ideals of linear type, have been intensely studied so far.
In this dissertation, we tackle the problem of describing the equations of R(I) for I =(J, y), with J being of linear type, i.e., for ideals of linear type up to one minimal generator. Throughout, such ideals will be referred to as ideals of almost-linear type. The main results of this work stem from two different approaches towards the problem.
In Theorem A, we give a full description of the equations of Rees algebras of ideals of the form I = (J,y), with J satisfying an homological vanishing condition. Theorem A permits us to recover and extend well-known results about families of ideals fulfilling the almost-linear type condition due to Vasconelos, Huckaba, Trung, Heinzer and Kim, among others.
Let a: S(I)¿R(I) be the canonical morphism from the symmetric algebra of I to the Rees algebra of I. In Theorem B, we prove that the injectivity of a single component of a: S(I)¿R(I) propagates downwards, provided I is of almost-linear type. In particular, this result gives a partial answer to a question posed by Tchernev. Finally, packs of examples are introduced, which illustrate the scope and applications of each of the results presented. The author also gives a collection of computations and examples which motivate ongoing and future research.